QGPINNs: A Physics-Informed Neural Network Framework for Nonlocal Differential Equations on Quantum Graphs

• URL: http://arxiv.org/abs/2608.28589v1
• Date: Fri, 28 Aug 2026 17:59:47 GMT
• Title: QGPINNs: A Physics-Informed Neural Network Framework for Nonlocal Differential Equations on Quantum Graphs
• Abstract summary: We propose QGPINNs, a physics-informed neural network framework for the numerical solution of nonlocal differential equations on quantum graphs.<n>The framework is designed as a general computational implementation in which the solution on each edge of the graph is approximated by a neural network.<n>We validate the accuracy, computational efficiency, and physical consistency of the proposed framework through numerical experiments on benchmark graph structures and real-world networks.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We propose QGPINNs, a physics-informed neural network framework developed in PyTorch for the numerical solution of nonlocal differential equations on quantum graphs. The framework is designed as a general computational implementation in which the solution on each edge of the graph is approximated by a neural network, while a unified graph-based loss function enforces the governing equations together with initial, boundary, and vertex transmission conditions. In particular, the formulation incorporates standard continuity and Kirchhoff-Neumann vertex conditions and Dirichlet boundary conditions into the learning process to couple the local edge-wise neural approximations into a global solution on the graph. The framework is developed for two representative classes of nonlinear models: multi-order fractional elliptic problems and time-fractional evolution equations on quantum graphs. To improve accuracy and training stability, QGPINNs integrates several graph-adapted learning strategies, including soft and hard constraint enforcement, dynamic loss balancing, Fourier feature embeddings, and a learnable singularity-capturing feature for weakly singular solutions arising in the considered problems. The framework also extends naturally to inverse problems, including the identification of the orders of fractional operators and physical parameters from noisy observational data. We validate the accuracy, computational efficiency, and physical consistency of the proposed framework through numerical experiments on benchmark graph structures and real-world networks, including the IEEE 14-bus system and an open-channel agricultural drainage network.

Related papers

• NewPINNs: Physics-Informing Neural Networks Using Conventional Solvers for Partial Differential Equations [6.108807911620144]
  We introduce NewPINNs, a physics-informing learning framework that couples neural networks with conventional numerical solvers.<n>NewPINNs integrates the solver directly into the training loop and defines learning objectives through solver-consistency.<n>We demonstrate the effectiveness of the proposed approach across multiple forward and inverse problems involving finite volume, finite element, and spectral solvers.
  arXiv  Detail & Related papers  (2026-01-23T22:34:57Z)
• DInf-Grid: A Neural Differential Equation Solver with Differentiable Feature Grids [73.28614344779076]
  We present a differentiable grid-based representation for efficiently solving differential equations (DEs)<n>Our results demonstrate a 5-20x speed-up over coordinate-based methods, solving differential equations in seconds or minutes while maintaining comparable accuracy and compactness.
  arXiv  Detail & Related papers  (2026-01-15T18:59:57Z)
• Graph Neural Regularizers for PDE Inverse Problems [62.49743146797144]
  We present a framework for solving a broad class of ill-posed inverse problems governed by partial differential equations (PDEs)<n>The forward problem is numerically solved using the finite element method (FEM)<n>We employ physics-inspired graph neural networks as learned regularizers, providing a robust, interpretable, and generalizable alternative to standard approaches.
  arXiv  Detail & Related papers  (2025-10-23T21:43:25Z)
• DFA-GNN: Forward Learning of Graph Neural Networks by Direct Feedback Alignment [57.62885438406724]
  Graph neural networks are recognized for their strong performance across various applications. BP has limitations that challenge its biological plausibility and affect the efficiency, scalability and parallelism of training neural networks for graph-based tasks. We propose DFA-GNN, a novel forward learning framework tailored for GNNs with a case study of semi-supervised learning.
  arXiv  Detail & Related papers  (2024-06-04T07:24:51Z)
• Implicit Stochastic Gradient Descent for Training Physics-informed Neural Networks [51.92362217307946]
  Physics-informed neural networks (PINNs) have effectively been demonstrated in solving forward and inverse differential equation problems. PINNs are trapped in training failures when the target functions to be approximated exhibit high-frequency or multi-scale features. In this paper, we propose to employ implicit gradient descent (ISGD) method to train PINNs for improving the stability of training process.
  arXiv  Detail & Related papers  (2023-03-03T08:17:47Z)
• Physics-aware deep learning framework for linear elasticity [0.0]
  The paper presents an efficient and robust data-driven deep learning (DL) computational framework for linear continuum elasticity problems. For an accurate representation of the field variables, a multi-objective loss function is proposed. Several benchmark problems including the Airimaty solution to elasticity and the Kirchhoff-Love plate problem are solved.
  arXiv  Detail & Related papers  (2023-02-19T20:33:32Z)
• Tunable Complexity Benchmarks for Evaluating Physics-Informed Neural Networks on Coupled Ordinary Differential Equations [64.78260098263489]
  In this work, we assess the ability of physics-informed neural networks (PINNs) to solve increasingly-complex coupled ordinary differential equations (ODEs) We show that PINNs eventually fail to produce correct solutions to these benchmarks as their complexity increases. We identify several reasons why this may be the case, including insufficient network capacity, poor conditioning of the ODEs, and high local curvature, as measured by the Laplacian of the PINN loss.
  arXiv  Detail & Related papers  (2022-10-14T15:01:32Z)
• A deep learning approach to solve forward differential problems on graphs [6.756351172952362]
  We propose a novel deep learning approach to solve one-dimensional non-linear elliptic, parabolic, and hyperbolic problems on graphs. A system of physics-informed neural network (PINN) models is used to solve the differential equations.
  arXiv  Detail & Related papers  (2022-10-07T16:06:42Z)
• Physics-constrained Unsupervised Learning of Partial Differential Equations using Meshes [1.066048003460524]
  Graph neural networks show promise in accurately representing irregularly meshed objects and learning their dynamics. In this work, we represent meshes naturally as graphs, process these using Graph Networks, and formulate our physics-based loss to provide an unsupervised learning framework for partial differential equations (PDE) Our framework will enable the application of PDE solvers in interactive settings, such as model-based control of soft-body deformations.
  arXiv  Detail & Related papers  (2022-03-30T19:22:56Z)
• Message Passing Neural PDE Solvers [60.77761603258397]
  We build a neural message passing solver, replacing allally designed components in the graph with backprop-optimized neural function approximators. We show that neural message passing solvers representationally contain some classical methods, such as finite differences, finite volumes, and WENO schemes. We validate our method on various fluid-like flow problems, demonstrating fast, stable, and accurate performance across different domain topologies, equation parameters, discretizations, etc., in 1D and 2D.
  arXiv  Detail & Related papers  (2022-02-07T17:47:46Z)
• Graph-Coupled Oscillator Networks [23.597444325599835]
  Graph-Coupled Networks (GraphCON) is a novel framework for deep learning on graphs. We show that our framework offers competitive performance with respect to the state-of-the-art on a variety of graph-based learning tasks.
  arXiv  Detail & Related papers  (2022-02-04T18:29:49Z)
• Multipole Graph Neural Operator for Parametric Partial Differential Equations [57.90284928158383]
  One of the main challenges in using deep learning-based methods for simulating physical systems is formulating physics-based data. We propose a novel multi-level graph neural network framework that captures interaction at all ranges with only linear complexity. Experiments confirm our multi-graph network learns discretization-invariant solution operators to PDEs and can be evaluated in linear time.
  arXiv  Detail & Related papers  (2020-06-16T21:56:22Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.